World Library  
Flag as Inappropriate
Email this Article

Factorial moment generating function

Article Id: WHEBN0004389572
Reproduction Date:

Title: Factorial moment generating function  
Author: World Heritage Encyclopedia
Language: English
Subject: Catalog of articles in probability theory, Generating functions, List of probability topics, List of statistics articles
Collection: Factorial and Binomial Topics, Generating Functions, Theory of Probability Distributions
Publisher: World Heritage Encyclopedia
Publication
Date:
 

Factorial moment generating function

In probability theory and statistics, the factorial moment generating function of the probability distribution of a real-valued random variable X is defined as

M_X(t)=\operatorname{E}\bigl[t^{X}\bigr]

for all complex numbers t for which this expected value exists. This is the case at least for all t on the unit circle |t|=1, see characteristic function. If X is a discrete random variable taking values only in the set {0,1, ...} of non-negative integers, then M_X is also called probability-generating function of X and M_X(t) is well-defined at least for all t on the closed unit disk |t|\le1.

The factorial moment generating function generates the factorial moments of the probability distribution. Provided M_X exists in a neighbourhood of t = 1, the nth factorial moment is given by [1]

\operatorname{E}[(X)_n]=M_X^{(n)}(1)=\left.\frac{\mathrm{d}^n}{\mathrm{d}t^n}\right|_{t=1} M_X(t),

where the Pochhammer symbol (x)n is the falling factorial

(x)_n = x(x-1)(x-2)\cdots(x-n+1).\,

(Many mathematicians, especially in the field of special functions, use the same notation to represent the rising factorial.)

Example

Suppose X has a Poisson distribution with expected value λ, then its factorial moment generating function is

M_X(t) =\sum_{k=0}^\infty t^k\underbrace{\operatorname{P}(X=k)}_{=\,\lambda^ke^{-\lambda}/k!} =e^{-\lambda}\sum_{k=0}^\infty \frac{(t\lambda)^k}{k!} = e^{\lambda(t-1)},\qquad t\in\mathbb{C},

(use the definition of the exponential function) and thus we have

\operatorname{E}[(X)_n]=\lambda^n.

See also

  1. ^ http://homepages.nyu.edu/~bpn207/Teaching/2005/Stat/Generating_Functions.pdf
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
 
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
 
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.
 


Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.